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This question got answered by Gjergji Zaimi and Richard Stanley in the comments. I simply reproduce their comments here as an answer:

A very simple explanation for this identity comes from the theory of symmetric functions. The ring $\Lambda$ of symmetric functions in infinitely many variables comes with an involution $\omega$, which interchanges the complete symmetric function $h_\lambda$ with the elementary symmetric function $e_\lambda$ for each partition $\lambda$.

Comparing the answers obtained for the trace of $\omega$ on homogeneous symmetric functions of degree $n$ using Schur functions and power sum symmetric functions yields the identity in question, for $\omega(s_\lambda)=s_{\lambda'}$ (giving trace as the number of self-transpose partitions) and $\omega(p_\lambda)=\epsilon(\lambda)p_\lambda$, where $\epsilon(\lambda)$ is the sign of a permutation with cycle decomposition $\lambda$ (giving trace as number of even partitions minus number of odd partitions).

Proofs of these facts concerning symmetric functions can be found in Stanley's Enumerative Combinatorics 2 (Sections 7.7 and 7.14).

A bijective proof for this identity was given by Marc van Leeuwen to the same question on http://math.stackexchange.com/a/102293/10126; he constructs an explicit bijection between the sets of even and odd partitions which do not have distinct odd parts. Also, there is a fairly standard bijection between partitions with distinct odd parts and self-transpose partitions.